Algebra Examples

$\frac{\sqrt{3 x^{3}}}{\sqrt{5 x}} \cdot \sqrt{15 x^{4}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $3 x^{3}$ as $x^{2} \cdot (3 x)$ .

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Step 1.1.1

Factor out $x^{2}$ .

$\frac{\sqrt{3 (x^{2} x)}}{\sqrt{5 x}} \cdot \sqrt{15 x^{4}}$

Step 1.1.2

Reorder $3$ and $x^{2}$ .

$\frac{\sqrt{x^{2} \cdot 3 x}}{\sqrt{5 x}} \cdot \sqrt{15 x^{4}}$

Step 1.1.3

Add parentheses.

$\frac{\sqrt{x^{2} \cdot (3 x)}}{\sqrt{5 x}} \cdot \sqrt{15 x^{4}}$

Step 1.2

Pull terms out from under the radical.

$\frac{x \sqrt{3 x}}{\sqrt{5 x}} \cdot \sqrt{15 x^{4}}$

Step 2

Combine $\sqrt{3 x}$ and $\sqrt{5 x}$ into a single radical.

$x \sqrt{\frac{3 x}{5 x}} \cdot \sqrt{15 x^{4}}$

Step 3

Reduce the expression $\frac{3 x}{5 x}$ by cancelling the common factors.

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Step 3.1

Cancel the common factor.

$x \sqrt{\frac{3 x}{5 x}} \cdot \sqrt{15 x^{4}}$

Step 3.2

Rewrite the expression.

$x \sqrt{\frac{3}{5}} \cdot \sqrt{15 x^{4}}$

Step 4

Rewrite $\sqrt{\frac{3}{5}}$ as $\frac{\sqrt{3}}{\sqrt{5}}$ .

$x \frac{\sqrt{3}}{\sqrt{5}} \cdot \sqrt{15 x^{4}}$

Step 5

Multiply $\frac{\sqrt{3}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x (\frac{\sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}) \cdot \sqrt{15 x^{4}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{\sqrt{3}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x \frac{\sqrt{3} \sqrt{5}}{\sqrt{5} \sqrt{5}} \cdot \sqrt{15 x^{4}}$

Step 6.2

Raise $\sqrt{5}$ to the power of $1$ .

$x \frac{\sqrt{3} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}} \cdot \sqrt{15 x^{4}}$

Step 6.3

Raise $\sqrt{5}$ to the power of $1$ .

$x \frac{\sqrt{3} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} \cdot \sqrt{15 x^{4}}$

Step 6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x \frac{\sqrt{3} \sqrt{5}}{{\sqrt{5}}^{1 + 1}} \cdot \sqrt{15 x^{4}}$

Step 6.5

Add $1$ and $1$ .

$x \frac{\sqrt{3} \sqrt{5}}{{\sqrt{5}}^{2}} \cdot \sqrt{15 x^{4}}$

Step 6.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x \frac{\sqrt{3} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}} \cdot \sqrt{15 x^{4}}$

Step 6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x \frac{\sqrt{3} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}} \cdot \sqrt{15 x^{4}}$

Step 6.6.3

Combine $\frac{1}{2}$ and $2$ .

$x \frac{\sqrt{3} \sqrt{5}}{5^{\frac{2}{2}}} \cdot \sqrt{15 x^{4}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$x \frac{\sqrt{3} \sqrt{5}}{5^{\frac{2}{2}}} \cdot \sqrt{15 x^{4}}$

Step 6.6.4.2

Rewrite the expression.

$x \frac{\sqrt{3} \sqrt{5}}{5^{1}} \cdot \sqrt{15 x^{4}}$

Step 6.6.5

Evaluate the exponent.

$x \frac{\sqrt{3} \sqrt{5}}{5} \cdot \sqrt{15 x^{4}}$

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

$x \frac{\sqrt{3 \cdot 5}}{5} \cdot \sqrt{15 x^{4}}$

Step 7.2

Multiply $3$ by $5$ .

$x \frac{\sqrt{15}}{5} \cdot \sqrt{15 x^{4}}$

Step 8

Combine $x$ and $\frac{\sqrt{15}}{5}$ .

$\frac{x \sqrt{15}}{5} \cdot \sqrt{15 x^{4}}$

Step 9

Rewrite $15 x^{4}$ as ${(x^{2})}^{2} \cdot 15$ .

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Step 9.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{x \sqrt{15}}{5} \cdot \sqrt{15 {(x^{2})}^{2}}$

Step 9.2

Reorder $15$ and ${(x^{2})}^{2}$ .

$\frac{x \sqrt{15}}{5} \cdot \sqrt{{(x^{2})}^{2} \cdot 15}$

Step 10

Pull terms out from under the radical.

$\frac{x \sqrt{15}}{5} \cdot (x^{2} \sqrt{15})$

Step 11

Rewrite using the commutative property of multiplication.

$\sqrt{15} \frac{x \sqrt{15}}{5} x^{2}$

Step 12

Multiply $\sqrt{15} \frac{x \sqrt{15}}{5}$ .

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Step 12.1

Combine $\sqrt{15}$ and $\frac{x \sqrt{15}}{5}$ .

$\frac{\sqrt{15} (x \sqrt{15})}{5} x^{2}$

Step 12.2

Raise $\sqrt{15}$ to the power of $1$ .

$\frac{x ({\sqrt{15}}^{1} \sqrt{15})}{5} x^{2}$

Step 12.3

Raise $\sqrt{15}$ to the power of $1$ .

$\frac{x ({\sqrt{15}}^{1} {\sqrt{15}}^{1})}{5} x^{2}$

Step 12.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x {\sqrt{15}}^{1 + 1}}{5} x^{2}$

Step 12.5

Add $1$ and $1$ .

$\frac{x {\sqrt{15}}^{2}}{5} x^{2}$

Step 13

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 13.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\frac{x {(15^{\frac{1}{2}})}^{2}}{5} x^{2}$

Step 13.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x \cdot 15^{\frac{1}{2} \cdot 2}}{5} x^{2}$

Step 13.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{x \cdot 15^{\frac{2}{2}}}{5} x^{2}$

Step 13.4

Cancel the common factor of $2$ .

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Step 13.4.1

Cancel the common factor.

$\frac{x \cdot 15^{\frac{2}{2}}}{5} x^{2}$

Step 13.4.2

Rewrite the expression.

$\frac{x \cdot 15^{1}}{5} x^{2}$

Step 13.5

Evaluate the exponent.

$\frac{x \cdot 15}{5} x^{2}$

Step 14

Cancel the common factor of $15$ and $5$ .

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Step 14.1

Factor $5$ out of $x \cdot 15$ .

$\frac{5 (x \cdot 3)}{5} x^{2}$

Step 14.2

Cancel the common factors.

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Step 14.2.1

Factor $5$ out of $5$ .

$\frac{5 (x \cdot 3)}{5 (1)} x^{2}$

Step 14.2.2

Cancel the common factor.

$\frac{5 (x \cdot 3)}{5 \cdot 1} x^{2}$

Step 14.2.3

Rewrite the expression.

$\frac{x \cdot 3}{1} x^{2}$

Step 14.2.4

Divide $x \cdot 3$ by $1$ .

$x \cdot 3 x^{2}$

Step 15

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 15.1

Move $x^{2}$ .

$x^{2} x \cdot 3$

Step 15.2

Multiply $x^{2}$ by $x$ .

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Step 15.2.1

Raise $x$ to the power of $1$ .

$x^{2} x^{1} \cdot 3$

Step 15.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 + 1} \cdot 3$

Step 15.3

Add $2$ and $1$ .

$x^{3} \cdot 3$

Step 16

Move $3$ to the left of $x^{3}$ .

$3 x^{3}$